game theoryh @ q ch 81 introduction what economists call game theory psychologists call the theory...

GAME THEORY H @ Q CH 8 1

Introduction What economists call game theory psychologists call the theory of social situations,

which is an accurate description of what game theory is about. Although game theory is relevant to parlor games such as poker or bridge, most research in game theory focuses on how groups of people interact. There are two main branches of game theory: cooperative and noncooperative game theory. Noncooperative game theory deals largely with how intelligent individuals interact with one another in an effort to achieve their own goals

In addition to game theory, economic theory has three other main branches: decision theory, general equilibrium theory and mechanism design theory. All are closely connected to game theory.

Decision theory can be viewed as a theory of one person games, or a game of a single player against nature. The focus is on preferences and the formation of beliefs.

General equilibrium theory can be viewed as a specialized branch of game theory that deals with trade and production, and typically with a relatively large number of individual consumers and producers

Mechanism design theory differs from game theory in that game theory takes the rules of the game as given, while mechanism design theory asks about the consequences of different types of rules. Naturally this relies heavily on game theory.

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http://levine.sscnet.ucla.edu/general/whatis.htm#Mechanism%20design%20theory


Two person Zero-sum gameNon cooperative game

1- a game consist of sequence of move ( like chess) or single move (present case).

2- strategy is the specification of a particular move for each of the players .

3- games can be classified on the basis of two criteria ; number of participants and net outcome of the game ;

3-a- one person zero-sum game ; player gets nothing

3-b- one person non-zero-sum game ; monopolist , monopsonist .

3-c- two person non-zero-sum game ; duopolist .

3-d- two person zero-sum game ; strictly competitive , no room for cooperation . Non-cooperative game

Two person zero-sum game ;

Suppose that the player I and player II has n strategy which has been identified as follows which is called the payoff matrix ;

3

Two person Zero-sum Two person Zero-sum Non cooperativeNon cooperative game game I’ s profit matrix ( equal to II’ s loss) ; player’s II strategy

a11 a12 a13 …..a1n

player’s I strategy a21 a22 a23 …..a2n …………………… an1 an2 an3 …..ann

aij : I’ s profit (or II’s loss ) when I employs his i th strategy (row) and II employs his j th strategy ( column) . (i denotes rows and j denoted columns ), suppose the payoff matrix is equal to ; a11=8 a12= 40 a13=20 a14=5 a21=10 a22= 30 a23=-10 a24=-8As it is seen this is a two players zero-sum single move game. For example if I chooses the first strategy and II chooses his first strategy , I’s profit is 8 , and II’ s profit is -8 ( or II’s loss is 8 ). I fears that II might discover his choice of strategy and desires to play it safe. If player I selects his ith strategy (ith row in the matrix) , his minimum profit and hence II’s maximum profit is given by the smallest element in the ith row of the pay-off matrix (min aij). I desires to maximize his minimum anticipated profit . Therefore I selects the strategy i for which min aij is the greatest , because he knows that for each strategy which he chooses ( each row) II will choose the smallest number in that row . player I strategy is Max Min aij .


Two person Zero-sum gamePlayer II possesses the same fear regarding I’s information and behavior. If II employs his jth strategy (jth column ) . He fears that I may employ the strategy corresponding to the largest element in the jth column of the payoff matrix , or max aij .Therefore II selects the strategy j for which Max aij is the smallest. In other words player II strategy is Min Max aij .the decision of these two player or are consistent or the equilibrium is achieved if ((player I strategy) ) Min Max aij= Max Min aij (player II strategy)

Regarding the example we will see that ; Min Max aij = Max Min aij =a14 = 5 So I’s first and II’s fourth strategy is the equilibrium pair of strategy. Neither duopolistic can increase profit by changing his strategy if his opponent remains unchanged . Now assume the following profit matrix of I’s profit ; -2 4 -1 6 3 -1 5 10Player II will never employ his third strategy , since he can always do better by employing his first strategy regardless of the I’s strategy. So first strategy dominates the third strategy .


Two person Zero-sum game

For player II ; j th column dominates kth column if aij ≤ aik for all i and for at least one i aij < aik . As is seen in the example the fourth column is dominated by first , second , and third column , and third column is dominated by first column .

For player I ; I th row dominates the h th row if aij≥ahj for all j , and aij > ahj for at least one j . As it is seen in the example neither of the rows is dominated the other one .eliminating all the columns which are dominated by others we will get the following payoff matrix ;

player II strategy

player I -2(a11 ) 4(a12)

strategy 3 (a21 ) -1(a22 )

As it is seen , equilibrium does not exist . Since ;

(player II ) Min Max aij =a21 =3 ≠ (player I ) Max Min aij player =a22 =-1


Two person Zero-sum gameMixed strategies ;A particular game as described above may or may not have solution if players select their strategy in the manner described above . This could be solved if players (duopolists) select their strategy on a probabilistic base .

Let r1…..rm be the probabilities with which player I will employ each of the m possible strategies ; 0 ≤ ri ≤ 1 , i=1 , 2 ,3 ….m , Σi

mri=1. a random selection will not allow player II to anticipate player I move even if he knows I’s probabilities .

player II can randomize his strategy by assigning the probabilities s1, s2 ,,,,sn to his strategy ; 0 ≤sj≤1 , j=1,2,3,,,n , Σj

nsj=1 .Players (duopolists) are now concerned with expected rather than actual concerned with expected rather than actual profits.profits. The decision problem of each player (duopolist) is to select an optimal set of probabilities. The probabilities are defined optimal if The probabilities are defined optimal if

player I gain= Σim aij ri ≥ v, for each strategy of II (j=1,2,..n)

v=value of game

player II loss= Σjn aij sj ≤ v, for each strategy of I (i=1,2,3,,,m)

v=value of game



I’s expected profit is at least as great as I’s expected profit is at least as great as v if II employs any of his pure strategies j with certainity (probability =1 ).

Player II’s expected loss is at least as small as Player II’s expected loss is at least as small as v if I employs any of his pure strategies i with certainity (probability =1 ) .

Fundamental theorems of game theories suggest that solution does exist for the probabilities r and s .

Player I expected profit = E1 = Σj=1nsj(Σi=1

maijri) ≥ Σj=1nsj v , Σj=1

nsj =1

EE11 = = ΣΣj=1j=1nn ΣΣi=1i=1

mm a aijij r ri i ssjj ≥ v ≥ v

Player II expected loss = E2 = Σi=1m ri (Σj=1

n aij sj) ≤ Σi=1m ri v , Σi=1

m ri =1

EE22 = = ΣΣj=1j=1nn ΣΣi=1i=1

mm a aijij r ri i ssjj ≤ v ≤ v

E1 ≥ v , E2 ≤ v , →→ E1 = E2 = V

As it could be seen the expected total outcome should be same for each of the players (duopolist) and equal to the value of the game if both players employ their optimal probabilities .



Linear programming equivalence Suppose that ZZjj = s = sj j / v/ v , Σj=1

nsj =1 , ZZ11 + Z + Z22 + Z + Z33 +….. Z +….. Znn = 1/v = 1/v , ,Player II desires to make his expected loss ( v ) as small as possible.Player II desires to make his expected loss ( v ) as small as possible. Or Max Max 1/v = Z1/v = Z11 + Z + Z22 + Z + Z33 +….. Z +….. Zn n

S.T. S.T. ΣΣj=1j=1nn a aij ij ZZj j ≤ 1 →→ ≤ 1 →→ aai1i1 Z Z11 + a + ai2 i2 ZZ22 + a + ai3 i3 ZZ33 +…..a +…..aininZZn n ≤ 1≤ 1

( since ; Σj=1n aij sj ≤ v , or Σj=1

n aij (sj /v) ≤ 1 , i=1,2,3,,,,m , Zj≥0) Also define wwi i = r= rii/v/v , Σi=1

m ri =1 → ww11 + w + w22 + w + w33 +…+w +…+wn n =1/v=1/vPlayer I desires to make his expected profit (v) as large as possible , so ; Min Min 1/v = w1/v = w11 + w + w22 + w + w33 +…..w +…..wnn

S.T. S.T. ΣΣi=1i=1mmaaij ij (r(ri i /v) ≥ 1 →→ /v) ≥ 1 →→ aa1j1j w w1 1 + a+ a2j 2j ww2 2 +a+a3j 3j ww33…… +a…… +amjmjwwmm ≥ 1 ≥ 1

( since ; Σi=1maijri ≥ v , or Σi=1

maij (ri /v) ≥ 1 , j= 1,2,3,4…n ) wi≥0The linear programming formulation facilitate a proof that solution always solution always exist for the two person zero-sum gameexist for the two person zero-sum game h.the proof proceeds by first establishing that finite optimal solution always exist for the equivalent programming system , and by then demonstrating that the optimal programming solution provide a solution for underlying game.


Two person Two person Nonzero-sum gameNonzero-sum game

Players (duopolists) are not always opposed . Their behavior may be characterized by a combination of competitive or cooperative elements or behavior.

The The possibility of cooperation arises in nonzero-sum gamepossibility of cooperation arises in nonzero-sum game . Such games do not necessarily leads to cooperation , although preferred outcomes may be achieved through cooperation.

Consider a simple duopolistic modelduopolistic model for which the collusion solution is prohibited by law . Bribe and profit redistribution is also illegal . Each hasEach has two strategytwo strategy ; ;

1- declare himself a declare himself a leaderleader and produce a large profit , or

2- declare himself a declare himself a followerfollower and produce a small profit .

once each player declare himself a position , he must produce the declared output regardless of what the rival has declared.

The matrix payoff for duopolist profit is as follows ;


Two person Nonzero-sum game PLAYER II

Leader Follower

PLAYER I Leader 200 250 1000 200

Follower 150 950 800 800

Sensible strategy (outcome) for each would be to declare himself a follower , since each would receive a moderately satisfactory profit . However, if I believes that II will declare himself a follower , he would be leader. The same is true for player II .

In fact the leaderleader strategy is the strategy is the dominantdominant strategy strategy for both of the players , since whatever strategy each would choose the other one select leader strategy for sure.

Since each has an incentive to be a leader , their uncooperativeuncooperative behavior leads to attain the lowest profit level.

It is clear that both would gain from cooperative behavior, but it is not clear that cooperation can be achieved successfully . Even if each agrees to be a follower , each would have an incentive to break the agreementincentive to break the agreement and declare himself a leader.


Two person Nonzero-sum game

Prisoners DilemmaPrisoners Dilemma Recent developments in game theory, especially the award of the Nobel Memorial Prize in 1994 to three game theorists and the death of A. W. Tucker, in January, 1995, at 89, have renewed the memory of its beginnings. Although the history of game theory can be traced back earlier, the key period for the emergence of game theory was the decade of the 1940's. The publication of The Theory of Games and Economic Behavior was a particularly important step, of course. But in some ways, Tucker's invention of the Prisoners' Dilemma example was even more important. This example, which can be set out in one page, could be the most influential one page in the social sciences in the latter half of the twentieth century.

This remarkable innovation did not come out in a research paper, but in a classroom. As S. J. Hagenmayer wrote in the Philadelphia Inquirer ("Albert W. Tucker, 89, Famed Mathematician," Thursday, Feb. 2, 1995, p.. B7) " In 1950, while while addressing an audience of psychologists at Stanford University, where he was a addressing an audience of psychologists at Stanford University, where he was a visiting professor, Mr. Tucker created the Prisoners' Dilemma to illustrate the visiting professor, Mr. Tucker created the Prisoners' Dilemma to illustrate the difficulty of analyzing" certain kinds of gamesdifficulty of analyzing" certain kinds of games. "Mr. Tucker's simple explanation has since given rise to a vast body of literature in subjects as diverse as philosophy, ethics, biology, sociology, political science, economics, and, of course, game theory."



Prisoners Dilemma Tucker began with a little story, like this: two burglars, Bob and Al, are captured near the scene of a burglary and are given the "third degree" separately by the police. Each has to choose whether or not to confess and implicate the other. If neither man confesses, then both will serve one year on a charge of carrying a concealed weapon. If each confesses and implicates the other, both will go to prison for 10 years. However, if one burglar confesses and implicates the other, and the other burglar does not confess, the one who has collaborated with the police will go free, while the other burglar will go to prison for 20 years on the maximum charge.

The strategies in this case are: confess or don't confess. The payoffs (penalties, actually) are the sentences served. We can express all this compactly in a "payoff table" of a kind that has become pretty standard in game theory. Here is the payoff table for the Prisoners' Dilemma game:



Prisoners Dilemma AL

CONFESS DON’T

CONFESS 10 10 0 20

BOB DONT 20 0 1 1

The table is read like this: Each prisoner chooses one of the two strategies. In effect, Al chooses a column and Bob chooses a row. The two numbers in each cell tell the outcomes for the two prisoners when the corresponding pair of strategies is chosen. The number to the left of the comma tells the payoff to the person who chooses the rows (Bob) while the number to the right of the column tells the payoff to the person who chooses the columns (Al). Thus (reading down the first column) if they both confess, each gets 10 years, but if Al confesses and Bob does not, Bob gets 20 and Al goes free



Prisoners Dilemma So: how to solve this game? What strategies are "rational" if both men want to minimize the time they spend in jail? Al might reason as follows: "Two things can happen: Bob can confess or Bob can keep quiet. Suppose Bob confesses. Then AI get 20 years if he doesn't confess, 10 years if he does, so in that case it's best to confess. On the other hand, if Bob doesn't confess, and AI doesn't either, AI get a year; but in that case, if AI confess he can go free. Either way, it's best if AI confess.

Therefore, AI confess.“ But Bob can and presumably will reason in the same way -- so that they both confess and go to prison for 10 years each. Yet, if if they had acted "they had acted "irrationally,"irrationally," and kept quiet, they each could have gotten and kept quiet, they each could have gotten off with one year each.off with one year each.



Prisoners Dilemma Dominant Strategies

What has happened here is that the two prisoners have fallen into

something called a "dominant strategy equilibrium."Dominant Strategy: Let an individual player in a game evaluate separately each of the strategy combinations he may face, and, for each combination, choose from his own strategies the one that gives the best payoff. If the If the same strategy is chosen for each of the different combinations of strategies same strategy is chosen for each of the different combinations of strategies the player might face, that strategy is called a dominant strategy" for that the player might face, that strategy is called a dominant strategy" for that player in that game.player in that game.

Dominant Strategy Equilibrium: If, in a game, each player has a dominant strategy, and each player plays the dominant strategy, then that combination of (dominant) strategiescombination of (dominant) strategies and the corresponding payoffs are said to constitute the dominant strategy equilibrium for that gameconstitute the dominant strategy equilibrium for that game. In the Prisoners' Dilemma game, to confess is a dominant strategy, and when both prisoners confess, that is a dominant strategy equilibrium.


firms A and B profits in two different strategies T > R > P > S and R > (T+S)/2 , if this cell is the solution.

Payoff Matrix : conditions for having Payoff Matrix : conditions for having prisoners dilemma prisoners dilemma Payoff Matrix : conditions for having Payoff Matrix : conditions for having prisoners dilemma prisoners dilemma

Firm B

Firm A

Cooperate

Defect

Cooperate

A: (R = 3)

B: (R = 3)

A: (S = 0)

B: (T = 5)

Defect A: (T = 5)

B: (S = 0)

A: (P = 1)

B: (P = 1)

This case is just like a duopoly situation in which the firms decide to This case is just like a duopoly situation in which the firms decide to cooperate and the one who cheat will get all the benefit .Each player cooperate and the one who cheat will get all the benefit .Each player will benefit if both choose cooperation , but the rational decision will benefit if both choose cooperation , but the rational decision making by the players makes them to select the defect strategy and making by the players makes them to select the defect strategy and get the lowest reward . The prisoners dilemma is presentget the lowest reward . The prisoners dilemma is present


Structure of the GameStructure of the Game

• If both players Defect on each other, each gets P (the Punishment payoff);

• If both players Cooperate with each other, each gets R (the Reward payoff);

• If one player Defects and the other Cooperates, the Defector gets T (the Temptation payoff), and the Cooperator gets S (the Sucker payoff);


Structure of the Game - Structure of the Game - Cont’dCont’d

T > R > P > S and R > (T+S)/2.– Taking into account slide no. 16 , these

inequalities rank the payoffs for cooperating and defecting.

– The condition of R > (T+S)/2 is important if the game is to be repeated h . It ensures that individuals are better off cooperating with each other than they would be by taking turns defecting on each other.


Structure of the Game - Structure of the Game - Cont’dCont’d

• Iterative PD vs. Single PD– Single instance games of PD have a

“rational” decision. Always defect, since defecting is a dominating strategy. However, with iterative PD always defecting is not optimal since an “irrational” choice of mutual cooperation will cause a net gain for both players. This leads to the “Problem of Suboptimization”


Deterministic Strategies for Deterministic Strategies for the Prisoner’s Dilemmathe Prisoner’s Dilemma

Tit for TatTit for Tat

Tit for Two TatTit for Two Tat


Tit for Tat (TFT)Tit for Tat (TFT)

• The action chosen is based on the The action chosen is based on the opponent’s last move.opponent’s last move.– Thereafter, always choose the opponent’s

last move as your next move.– On the first turn, the previous move cannot

be known, so always cooperate on the first move (with the hope of getting cooperation from

the rival).

H @ Q CH 8 22

Tit for Tat (TFT)Tit for Tat (TFT)

II , C acceptable by the rule ( 3 ,3 )1- I , C II , D not acceptable by the rule ( 0 , 5 )

II , C not acceptable by the rule ( 5 ,0 )2 – I , D II , D acceptable by the rule ( 1,1 )

Knowing the rules of the game, player I chooses C. Knowing the rules of the game, player I chooses C. So we can see that like simultaneous move So we can see that like simultaneous move defecting in not an optimal strategy defecting in not an optimal strategy

Tree diagram explanation payoff matrix slide no. 16 with iterative move: In slide no.16 there is simultaneous move , but in this Example the move is not simultaneous. Player one ( I ) moves first and Player two ( II) moves next.


Key Points Key Points of Tit for Tat

– NiceNice; it cooperates on the first move.– RegulatoryRegulatory; it punishes defection with

defection.– ForgivingForgiving; it continues cooperation after

cooperation by the opponent.– ClearClear; it is easy for opponent to guess the

next move, so mutual benefit is easier to attain.


Tit for Two Tat (TF2T)Tit for Two Tat (TF2T)

• Same as Tit for Tat, but requires two consecutive defections(cooperations) for a defection(cooperation) to be returned.– Cooperate on the first two moves hoping the

opponent to cooperate in his move .– If the opponent defects twice in a row, player

chooses defection as the next move.


After each of the players makes two moves game will finishes taking into account the rules of the game that two consecutive defections (cooperation) needed from opponent for a defection ( cooperation ) to be returned.

I , CI , D

I, CI , D

I , CI , D

I , CI , D

I , CI , D

II , C

II , D II , CII , D

I , C

I , D

Cooperation will last

Not acceptable

II , DII , C

Cooperation will last

Not acceptable

Two repetition game. Fore more repetition more arrows should be drawn


Tit for Two Tat (TF2T)Tit for Two Tat (TF2T)

II , C

I ,C

II , C II , D

1- I , D I, D II , C not acceptable

II , D II , D

I ,C II , C

II , D

I , D II , C not acceptable

II , D defect will last

I , CI , C

I , DI , D

II , CII , C Cooperation will lastCooperation will last


• Key Points of Tit for Two Tat

– When defection is the opponent’s first move, this strategy outperforms Tit for Tat

– Cooperating after the first defection Cooperating after the first defection could cause the opponent to cooperate could cause the opponent to cooperate also. Thus, in the long run, both players also. Thus, in the long run, both players benefit more points.benefit more points.


More on prisoners dilemma

There are two persons who have committed a crime of which there is no evidence. Police catches them and puts them in two separate cells. Because there is no evidence against the convicts, they cannot be proven guilty. So the police tries to use one against the other. Each Prisoner is given two options either to confess his crime or to deny it . If prisoner I confesses but prisoner II denies then the first prisoner serves as Testimony against the other and he gets no punishment, while the prisoner II gets full term of 10 yrs and vice versa. If both confess both get 5 years of imprisonment each as now police has evidence against both of them. If both deny the police has evidence against none, so maximum punishment that they can get is 1 year each. This can be represented in tabular form as.


PLAYER IIPLAYER II CONFESS CONFESS DENYDENY

CONFESSCONFESS 5 5 0 10 P= 0.5 PLAYER IPLAYER I DENYDENY 10 0 1 1 P= 0.5

P= 0.5 P=0.5 P= 0.5 P=0.5

• This is the standard representation of 2 player game. Each cell has two payoffs, one for each player. The first number in a cell is the penalty of player 1 and the second number is the penalty of player two. Each row represents a strategy for player 1 and each column represents a strategy for player 2. So the bottom right column means if Player 1 denies and Player 2 denies then penalty for player 1 is 1 year and that of player two is also 1 year.Now lets analyze the Game with player I 's perspective.


• He doesn't know if player II is going to confess or deny, but he wants to decrease his punishment. So he considers two cases.

• a) If player II confesses In this case confessing gives 5 years imprisonment while denying gives 10 years So its better to confess

• b) If player II denies In this case confessing gives only 1 years imprisonment while denying gives 1 years Again its better to confess

• So player I will like to confess if he is guilty.• Player II will argue on similar lines and will also

like to confess if guilty


• If player 1 assumes that player 2 would If player 1 assumes that player 2 would confess with probability 0.5 ,The expected confess with probability 0.5 ,The expected number of years he will be in prison ifnumber of years he will be in prison if he he confesses with probability 0.5 is confesses with probability 0.5 is 0.50.5 x 0.5 x ( 5 )+ 0.5 x ( 5 )+ 0.50.5 x 0.5 x (10)0.5 x (10) + 0.50.5 x 0.5 0.5 (1)+ 0.50.5 x 0.5 0.5 (0) = 4 years.

• If player II chooses ConfessConfess with probability 0.40.4 and Deny with probability 0.6 0.6 He assumes that player IIII would confessconfess with probability 0.50.5


• for player I • 0.4 x 0.5 x 5 +

( I confesses ) ( II confesses ) ( I gets 5 years )• 0.6 x 0.5 x 10 +

( I denies ) ( II confesses ) ( I gets 10 years )• 0.4 x 0.5 x 0 +

( I confesses ) ( II denies ) ( I gets 0 years )• 0.6 x 0.5 x 1

( I denies ) ( II denies ) ( I gets 1 year )• = 4.3 years

• We see that if player I is less likely to confess his penalty We see that if player I is less likely to confess his penalty increases increases


Now assume Player I confesses with probability q and assumes that player II would confess with probability p The expected number of years of prison for player I : 5 pq + 0 x q(1-p) + 10 x ( 1-q )p + 1.(1-q)(1-p) = qp - q(4p+1) years this is a decreasing function of q. so as a general rule So more likely player I is to confess less So more likely player I is to confess less punishment he will get irrespective of what player punishment he will get irrespective of what player II does . II does .


• Individual's behavior at a traffic intersectiontraffic intersection is also similar to prisoners dilemma. When a commuter arrives and faces a red light he/she has two options.a) Waita) Wait for light to turn Green for light to turn Green b) Jumpb) Jump the Red light the Red light

• Lets call the strategy a as Obeya as Obey and strategy b as b as Disobey.Disobey. There are two players in this game. First playerFirst player is the commutercommuter and All other peopleAll other people at that intersection can be considered as the second playersecond player in the game.

1- 1- If the commuter obeys and others also obey he will have If the commuter obeys and others also obey he will have to suffer delay of 'd' that is the time required for the red to suffer delay of 'd' that is the time required for the red light to turn greenlight to turn green. .

2- If he disobeys but others obey his delay is 0.2- If he disobeys but others obey his delay is 0.3- If he obeys but others disobey let additional delay is D3- If he obeys but others disobey let additional delay is D

( due to congestion ) over 'd' . ( due to congestion ) over 'd' . 4- If all disobey total delay is D4- If all disobey total delay is D • Writing as Standard penalty Matrix


PLAYER IIPLAYER II OBEY DISOBEY

PLAYERPLAYER OBEY d d + D I DISOBEY 0 D

This game is similar to prisoners dilemma of example 4. If we analyze like last case the best best option for option for the the commuter (player I )commuter (player I ) is to is to disobeydisobey irrespective of what others do. This is what we see at traffic This is what we see at traffic lights if there is no fine for jumping the traffic light.lights if there is no fine for jumping the traffic light. Now if we introduce finesintroduce fines i.e. if the commuter is disobeying he can be caught by the trafficcaught by the traffic police with probability pprobability p .They finefine imposed is equal to f.equal to f. Let the penalty be c(d,f,p) i.e. a function of delay (d), fine (f) and probability (p) of being caught.


PLAYER II

OBEY DISOBEYPLAYER OBEY c( d , 0,0 ) c ( d+D, 0, 0 ) I DISOBEY c( 0 ,f, p ) c (D ,p ,f )

if we simply define penalty as c (d,f,p) = d+pf the penalty matrix reduces to

PLAYER II OBEY DISOBEY

PLAYER OBEY d d+D I DISOBEY 0+fp D+pf

If we put the fine such that pf > dpf > d then we can see that obeying is the obeying is the bestbest strategy strategy


Two person Nonzero-sum game Prisoners Dilemma Issues With Respect to the Prisoners' Dilemma

This remarkable result -- that individually rational action results in both persons being made worse off in terms of their own self-interested purposes -- is what has made the wide impact in modern social science. For there are many interactions in the modern world that seem very much like that, from

arms races through road congestion and pollution to the

depletion of fisheries and the overexploitation of some

subsurface water resources. p These are all quite different interactions in detail, but are interactions in which (we suppose) individually rational action leads to inferior results for each person, and the Prisoners' Dilemma suggests something of what is going on in each of them. That is the source of its power. Having said that, we must also admit candidly that the Prisoners' Dilemma is a very simplified and abstract -- if you will, "unrealistic" -- conception of many of these interactions.

A number of critical issues can be raised with the Prisoners' Dilemma, and each of these issues has been the basis of a large scholarly literature:



Prisoners Dilemma The Prisoners' dilemma is a two-person game, but many of the applications of the idea are really many-person interactions.

We have assumed that there is no communication between the two prisoners. If they could communicate and commit themselves to coordinated strategies, we would expect a quite different outcome.

In the Prisoners' Dilemma, the two prisoners interact only once. Repetition of the interactions might lead to quite different results.

Compelling as the reasoning that leads to the dominant strategy equilibrium may be, it is not the only way this problem might be reasoned out. Perhaps it is not really the most rational answer after all .We will consider some of these points in what follows.



Prisoners Dilemma An Information Technology ExampleGame theory provides a promising approach to understanding strategic problems of all sorts, and the simplicity and power of the Prisoners' Dilemma and similar examples make them a natural starting point. But there will often be complications complications we must consider in a more complex and realistic application. Let's see how we might move from a simpler to a more move from a simpler to a more Realistic gameRealistic game model in a real-world example of strategic thinking:

choosing an information system.choosing an information system.For this example, the players will be a companycompany considering the choice of choice of a new internal e-mail a new internal e-mail , and a suppliersupplier who is considering producing it. The two choicestwo choices are to install a 1- technically advanced or 2- a more 1- technically advanced or 2- a more proven system with less functionalityproven system with less functionality. We'll assume that the more advanced system really does supply a lot more functionality, so that the payoffs to the two players, net of the user's payment to the supplier, payoffs to the two players, net of the user's payment to the supplier, are as shown in Tableare as shown in Table


Two person Nonzero-sum game USER (company)

ADVANCED PROVEN

ADVANCED 20 20 0 0

SUPPLIER

PROVEN 0 0 5 5

We see that both players can be better off, on net, if an advanced system is installed. (We are not claiming that that's always the case! We're just assuming it is in this particular decision). But the worst that can happen is for one player to commit to an advance system while the other player stays with the proven one. In that case there is no deal, and no payoffs for anyone. The problem is that the supplier and the user must have a compatible standard, in order to work together, and since the choice of a standard is a strategic choice, their strategies have to mesh.

Although it looks a lot like the Prisoners' Dilemma at first glance, this is a more complicated game. We'll take several complications in turn:


Two person Nonzero-sum gamewe see that there this game has no dominated strategiesno dominated strategies. The Looking at it carefully, best strategy for each participant depends on the strategy chosen by best strategy for each participant depends on the strategy chosen by the other participantthe other participant. Thus, we need a new concept of game-equilibrium, that will allow for that complication. When there are no dominant strategies, we often use an equilibrium conception called the NASH EQUILIBRIUM NASH EQUILIBRIUM named after Nobel Memorial Laureate John Nash. The Nash Equilibrium is a pretty simple idea: we have a Nash Equilibrium if each participant chooses the we have a Nash Equilibrium if each participant chooses the best strategy, given the strategybest strategy, given the strategy chosen by the other participantchosen by the other participant. In the example, if the user opts for the advanced system, then it is best for the supplier to do that too. So (Advanced, Advanced) is a Nash-equilibrium. But, hold on here! If the user chooses the proven system, it's best for the supplier to do that too. There are two Nash EquilibriumThere are two Nash Equilibrium Which one will be chosenWhich one will be chosen? It may seem easy enough to opt for the advanced systemadvanced system which is better all around, but if each participant believes that the other will stick with the proven but if each participant believes that the other will stick with the proven systemsystem -- being a bit of a stick in the mud, perhaps -- then it will be best for each player to choose the proven system -- and each will be right in assuming that the other one is a stick in the mud! This is a danger typical of a class of This is a danger typical of a class of games called coordination games games called coordination games -- and what we have learned is that the choice of compatible standards is a coordination game.



We have assumed that the payoffs are known and certain. In the real world, every strategic decision is risky every strategic decision is risky -- and a decision for the advanced system is likely to be riskier than a decision for the proven system. Thus, we would have to take into account the players' subjective attitudes toward risk, their risk aversion, to make the example fully realistic. We won't attempt to do that in this example, but we must keep it in mind.

The example assumes that payoffs are measured in money. Thus, we are not only leaving risk aversion out of the picture, but also any other leaving risk aversion out of the picture, but also any other subjective rewards and penalties that cannot be measured in moneysubjective rewards and penalties that cannot be measured in money.

Economists have ways of measuring subjective rewards in money terms -- and sometimes they work -- but, again, we are going to skip over that problem and assume that all rewards and penalties are measured in assume that all rewards and penalties are measured in moneymoney and are transferable from the user to the supplier and vice versa.


Two person Nonzero-sum gameReal choices of information systems are likely to involve more than two players, at least in the long run -- the user may choose among several suppliers, and suppliers may have many customers. That makes the coordination problem harder to solve.

Suppose, for example, that "beta"beta" is the advancedadvanced system and "VHS"VHS"" is the provenproven system, and suppose that about 90% of the market uses "VHS." Then "VHS" may take over the market take over the market from "beta" even though "beta" is the better system. Many economists, game theorists and others believe this is a main reason why certain technical standards gain dominance.)


Two person Nonzero-sum gameOn the other hand, the user and the supplier don't have to just sit back and wait to see what the other person does -- -- they can sit down and talk it out, and they can sit down and talk it out, and commit themselves to a contractcommit themselves to a contract. In fact, they have to do so, because the amount of payment amount of payment from the user to the supplier -- a strategic decision we have ignored until now -- also has to be agreed upon.has to be agreed upon.

In other words, unlike the Prisoners' Dilemmaunlike the Prisoners' Dilemma, this is , this is a cooperative game, not a noncooperative gamea cooperative game, not a noncooperative game.. which will make the problem of coordinating standards easier, and it needs to apply different different approach .approach .


Games with Multiple Nash Equilibria

Here is another example to try the Nash Equilibrium approach on. Two radio stations (WIRD and KOOL) have to choose formats for their broadcasts. There are three possible formats: Country-Western (CW), Industrial Music (IM) or all-news (AN). The audiences for the three formats are 50%, 30%, and 20%, respectively. If they choose the same formats they will split If they choose the same formats they will split the audience for that format equally, while if they choose different the audience for that format equally, while if they choose different formats, each will get the total audience for that formatformats, each will get the total audience for that format. Audience shares are proportionate to payoffs. The payoffs (audience shares) are in Table 6-1.

KOOLKOOL

CW IM AN

CW 25 25 5050 3030 50 20

WIRDWIRD IM 30 30 50 50 15 15 30 20

AN 20 50 20 30 10 10


Games with Multiple Nash EquilibriaGames with Multiple Nash Equilibria

It should be able to verify that this is a non-constant sum game, and that there are no dominant strategy equilibria. If we find the Nash Equilibria by If we find the Nash Equilibria by elimination, we find that there are two equilibrium elimination, we find that there are two equilibrium ;the upper middle cell

((CWCW(50),(50),IMIM(30)(30))) and the middle-left one ,(,(CWCW(30)(30),,IMIM(50)(50))) in both of which one station chooses CW (and gets a 50 market share) and the other chooses IM (and gets 30) . But it doesn't matter which station chooses which format.

It may seem that this makes little difference, since the total payoff is the same since the total payoff is the same in both cases, namely 80 both are efficient, in that there is no larger total payoff in both cases, namely 80 both are efficient, in that there is no larger total payoff than 80 than 80

There are multiple Nash Equilibria in which neither of these things is so, as we will see in some later examples. But even when they are both true, the multiplication of equilibrium multiplication of equilibrium creates a dangercreates a danger. . 11-The danger is that both -The danger is that both stations will choose the more profitable CW format -- and split the market, stations will choose the more profitable CW format -- and split the market, getting only 25 each! getting only 25 each! 22-Actually, there is an even -Actually, there is an even worse danger worse danger that each that each station might assume that the other station will choose CW, and each station might assume that the other station will choose CW, and each choose IM, splitting that market and leaving each with a market share of choose IM, splitting that market and leaving each with a market share of just 15just 15. . More generally, the problem for the players is to figure out which

equilibrium will in fact occur.


Games with Multiple Nash EquilibriaIn still other words, a game of this kind raises a "coordination problem:" how can the two stations coordinate their choices of strategies and avoid the danger of a mutually inferior outcome such as splitting the market? Games that present coordination problems are sometimes called coordination games. From a mathematical point of view, this multiplicity of equilibrium is a problem. For a "solution" to a "problem," we want one answer, not a family of answers. And many economists would also regard it as a problem that has to be solved by some restriction of the assumptions that would rule out the multiple equilibrium. But, from a social scientific point of view, there is another interpretation. Many social scientists believe that coordination problems are quite real and important aspects of human social life. From this point of view, we might say that multiple Nash equilibrium provide us with a possible "explanation" of coordination problems. That would be an important positive finding, not a problem!



Another source of a hint that could solve a coordination gamesolve a coordination game is social convention.convention. Here is a game in which social convention could be quite important. That game has a long name: "Which Which Side of the Road to DriveSide of the Road to Drive On?" On?" In Britain, we know, people drive on the left side of the road; in the US they drive on the right. In abstract, how do we choose which side to drivehow do we choose which side to drive on?on? There are two strategies: drive on the left side and drive on the right side. There are two possible outcomes: the two cars pass one another without incident or they crash.We arbitrarily assign a value of 1 each to passing We arbitrarily assign a value of 1 each to passing without problems and of -10 each to a crash.without problems and of -10 each to a crash.

MERCIDES MERCIDES

L RL R

L L 1 1 1 1 -10-10 -10-10

BUIKBUIK

R R -10 -10 -10-10 11 11



Verify that LL and RR are both Nash equilibria. Verify that LL and RR are both Nash equilibria. But, if we do not know which side to chooseBut, if we do not know which side to choose, there is some danger that we will choose LR or RL at random and crash. How can we know which side to choose? The answer is, of course, that for this coordination game we rely we rely on social convention.on social convention. Conversely, we know that in this game, social convention is very powerful and persistent, and no less so in the country where the solution is LL than in the country where it is RR.


Nash Equilibrium and the

Minimax Strategyin Zero Sum Games

Two teams are playing football.Two teams are playing football. The offense can choose from among four strategies, shown as rows in the table. The defense can choose from three strategies to stop the play. The payoffs are yards gained by the offense, or The payoffs are yards gained by the offense, or yards lost by the defense. yards lost by the defense. Each yard gained by the offense is a yard lost by the defenseEach yard gained by the offense is a yard lost by the defense.If Offense chooses the long pass and the Defense runs a Blitz then the Offense is thrown for a loss of two yards; this is the worst, or minimum, outcome from choosing Long Pass. The final column of the table shows all of these worst possible outcomes for Offense. Offense should make the best of these bad outcomes by choosing to run the Short Pass play.If Defense chooses to defend against the run then the best they can do is give up 2 yards, the worst is give up ten yards. If they defend against the pass then the worst they can do is give up 5.6 yards. The last row shows the maxima of these bad outcomes. Defense should make the best of a bad situation by Defense should make the best of a bad situation by choosing that strategy which gives up the least yards of all. They should choosing that strategy which gives up the least yards of all. They should choose to defend against the passchoose to defend against the pass..


Defense

Run PassPass Blitz Min=

Offense

Run 2 5 13 2

Short Short PassPass

6 5.65.6 10.5 5.65.6

Medium Pass

6 4.5 1 1

Long Pass

10 3 -2 -2

Max= 10 5.65.6 13

Min Max of defense = Min Max of defense = Max Min of offenseMax Min of offense = 5.6 = 5.6


In many game settings neither player has aneither player has a dominantdominant or aor a dominated strategydominated strategy. In these cases we need other solution approaches. Examine the game to the right. By now you should be familiar enough with analyzing games that it is immediately apparent that neither player has a dominant strategy, and since each player has only two strategies no dominant strategy means there can be no dominated strategy either. When we can't rely oncan't rely on either dominant or dominateddominated strategies our next approach is to look for a Nash equilibriumNash equilibrium..

http://www.econweb.com/Sample/Oligopoly/Nash-2.html


a Nash equilibrium is somewhat different from either dominant or dominated strategies. Remember that a dominant strategy is always the best strategy to use and a dominated strategy is never the best strategy to use. In order to understand the Nash understand the Nash equilibrium concept we have to remember what an equilibrium concept we have to remember what an equilibrium is.equilibrium is. In economics an equilibrium means that no one wishes to change equilibrium means that no one wishes to change their behavior their behavior as long as nothing else changesas long as nothing else changes. Remember that a simple demand and supply equilibrium means that consumers are purchasing their desired quantity demanded and suppliers are selling their desired quantity supplied and neither wants to change as long as the other doesn't. But, we also learned that a change on one side of the market (such as a change in income or a change in production costs) will lead to new desired quantities on both sides. In other words, equilibrium quantity demanded depends on where the , equilibrium quantity demanded depends on where the supply curve is and equilibrium quantity supplied depends on supply curve is and equilibrium quantity supplied depends on where demand is. where demand is.


The following is a simple definition: A Nash equilibrium existsA Nash equilibrium exists when each player is doing the best she can given what the other player is doing. In other words, neither wishes to change strategies so long as the other player neither wishes to change strategies so long as the other player doesn't. doesn't. Consider the payoff matrix we saw before, shown again to the right. If Firm 2 plays High Firm 1 will play Low. As it turns out, if Firm 1 plays Low Firm 2 will wish to play High. For Firm 1 Low is only best if Firm 2 plays High and for Firm 2 High is best only if Firm 1 plays Low. Firm 1 Firm 1 Low,Low, Firm 2 High Firm 2 High is a Nash equilibrium is a Nash equilibrium because each is doing the best possible given what the other is doing. Can you see that this game has two Nash equilibrium? Firm 1 HighFirm 1 High and and firm 2 Lowfirm 2 Low is another Nash equilibrium . is another Nash equilibrium .


consider the driving game to the right. In many courtiers drivers drive on the right hand side of the road, in other countries they drive on the left. But, no matter where you drive, it's best to do what But, no matter where you drive, it's best to do what the other drivers are doingthe other drivers are doing.. This simple payoff matrix illustrates this. In the US for example drivers use the right side, so you should too In the US for example drivers use the right side, so you should too in the US. This is a Nash equilibrium. in the US. This is a Nash equilibrium. If you thoughtIf you thought for some reason for some reason everyone was going to drive on the left tomorrow it would be smart everyone was going to drive on the left tomorrow it would be smart for you to do the samefor you to do the same. In England, on the other hand, people drive In England, on the other hand, people drive on the left so if you are driving in England you should drive on the left on the left so if you are driving in England you should drive on the left as well. as well.


There are two Nash equilibrium for the driving game, everyone drive on the Left side of the road or everyone drive on the Right side of the road. This is one problem with the Nash equilibrium as a solution approach. It's fairly common for games with no dominant strategies It's fairly common for games with no dominant strategies to have more than one to have more than one Nash equilibriumNash equilibrium. .




Let's look again at the first game we saw in this section. One way One way we can tell a particular outcome is not a Nash equilibrium is we can tell a particular outcome is not a Nash equilibrium is to ask if the game ended there would either play wish she to ask if the game ended there would either play wish she could change her play? could change her play? Consider the High, High square to the right. If the game were to be played again and if Firm 1 believed Firm 2 would again play High then Firm 1 would wish to change to Low, as indicated by the arrow. Likewise, if Firm 2 believed that Firm 1 would stick with High Firm 2 would wish to switch to Low. So, High, High can't be a Nash equilibrium since both players aren't doing the best possible given what the other player is doing




We can also quickly confirm that Low, Low isn't a Nash equilibrium. Again, as indicated by the arrows, neither firm is content with playing Low so long as they believe the other will continue to do so. Remeber that a Remeber that a Nash equilibriumNash equilibrium means that each player is means that each player is doing the best possible doing the best possible givengiven what the other players are doing. what the other players are doing. One way we can track down Nash equilibrium is to put in arrows, as we have here, wherever either player would change her strategy if the other player did not.



If we have all our arrows drawn correctly we see that there are no arrows pointing away from Firm 1 Low, Firm 2 High. As long as Firm 1 believes that Firm 2 will play High Firm 1 will stick with Low. AND, as long as Firm 2 believes Firm 1 will play Low Firm 2 will continue playing High. Since both players are doing the best they can given what the other player is doing Firm 1 Low, Firm 2 High is a Nash equilibrium. Our method of drawing arrows and finding outcomes with no arrows pointing away works. This means there is yet another Nash equilibrium.


Since no arrows point away from Firm 1 High, Firm 2 Low it must also be a Nash equilibrium. A quick check should convince you that neither player will wish to change its strategy so long as the other does not. Keep in mind that a Nash equilibrium requires that both Nash equilibrium requires that both players be simultaneously doing the best possible given what players be simultaneously doing the best possible given what the other player is doing. the other player is doing. So long as nothing changes neither will wish to change her strategy, like any other equilibrium.. When a Nash equilibrium is present we should predict it as the outcome of the game, but here we have two and no compelling way but here we have two and no compelling way to choose one. to choose one.


A serious problem with using Nash equilibrium as a solution approach is that it is quite common for games to have multiple Nash equilibria. In fact both the games we've seen so far have two Nash equilibria. Sometimes when games have two Sometimes when games have two Nash equilibriaNash equilibria one is more one is more convincing than the otherconvincing than the other. Let's consider the following game . First we can easily see that there is no dominant strategy. Let's try using our method of drawing arrows drawing arrows to see if we can locate any Nash equilibria. Let's start with Firm 1 High, Firm 2 Low since this was a Nash equilibrium in our last game.


If Firm 2 believed Firm 1 would stick with High Firm 2 would wish to switch to High, indicated by the arrow. Likewise, if Firm 1 thought Firm 2 would stick with Low it would switch to Low as shown by the other arrow. Firm 1 High, Firm 2 Low cannot possibly be a Nash equilibrium since at least one (both in this case) of the players would wish to change strategies unilaterally. In other words, they would change strategies even if the other did not, so clearly this is outcome doesn't fit any sort of equilibrium idea.


Now let's consider Firm 1 Low, Firm 2 High as a possible Nash equilibrium. We can easily see that, if Firm 1 believed Firm 2 would play High it would like to switch to High as well, hence the arrow. This is enough to tell us that this outcome isn't a Nash equilibrium, if even one player would change her strategy if the other did not the outcome can't be an equilibrium. As it turns out, if Firm 2 believed Firm 1 would play Low it would like to switch to High, as the arrow shows. This too would be sufficient evidence to convince us this isn't an equilibrium outcome.


With all the arrows drawn in we see that there are two possible outcomes from which no arrows point away. If both play High, neither will wish to change so long as the other doesn't. The same is true if both play Low. Once again we have a game with two Nash equilibria. In this case it's a little easier to predict an outcome. The payoffs are sufficiently better for both players playing High than for both playing Low that in this case this is our best prediction.


In any game without dominant strategies but for which there exists Nash equilibrium, our best solution is that a Nash equilibrium will be the outcome. In some games however with multiple equilibrium whose payoffs are similar this isn't always very clear guidance, but in some games, such as the last one we examined, it will be fairly clear which equilibrium is the best solution to predict. To be sure you understand Nash equilibrium, see if you can explain to yourself why

all dominant strategy equilibrium are Nash all dominant strategy equilibrium are Nash equilibrium, but not all Nash equilibrium are equilibrium, but not all Nash equilibrium are dominant strategy equilibriumdominant strategy equilibrium.


Don't Confess Confess

Prisoner 1

Prisoner 2 Prisoner 2

Don't Confess

ConfessDon't Confess Confess

1,1 15,0 0,15 5,5

This line represents a constraint on the information that prisoner 2 has available. While 2 moves second, he does not know what 1 has chosen.

Payoffs are: Prisoner 1 payoff, Prisoner 2 payoff.

prisoner 1 prisoner 2

Extensive Form Extensive Form


Solving Extensive Form Games

Thus far we have considered strategic form games that were presented in the form of two-way tables. Such representations are awkward for picturing games in which the players move sequentially. In order to represent sequential games we need to explore extensive form games. There are two graphical representations of extensive form games. Either representation is known as a game tree.


In the first panelIn the first panel we have a game tree representing a sequential game. The starting point of the game or root of the tree is at the left edge of the picture. Player 1's two moves are a or b and are represented as two branches coming off the root. At the ends of the branches for Player 1 are the decision points for Player 2. The relevant decision point for Player 2, in blue, depends on the observed choice made by Player 1. Player 2 can also choose between strategy a and strategy b. The payoffs, denoted (a) and (b), are at the terminal nodes of the game tree. When it is her turn to move Player 2 knows what choice has been made by Player 1. This is denoted by the fact that Player 2's branches are explicitly attached to the nodes of Player 1's branches.




Information set


The terminology of this game tree is the same as that for the previous game tree. There is one important difference. The The branches for Player 2's decisions are not attached directly branches for Player 2's decisions are not attached directly to the branches for Player 1. At the time player 2 makes to the branches for Player 1. At the time player 2 makes her decision she doesn't know what strategy has been her decision she doesn't know what strategy has been chosen by Player 1.chosen by Player 1. The ellipse is known as the The ellipse is known as the information setinformation set. Contained in the information set is an enumeration of the strategies available to Player 1 and the payoffs associated with those strategies. Also contained in the information set is an enumeration of the strategies available to Player 2 and the associated payoffs. In effect the information In effect the information set contains all of the data necessary for the players to set contains all of the data necessary for the players to make their decisions simultaneouslymake their decisions simultaneously



The conditions necessary to represent a game as a tree 1.1.A single starting point A single starting point

2.2.No cyclesNo cycles

3.3.One way to proceedOne way to proceed StrategiesStrategies:: A player's strategy is a complete conditional plancomplete conditional plan of action.Mixed StrategiesMixed Strategies: A mixed strategy is a probability distribution over the pure strategies, the support, that might be played.Chance NodesChance Nodes: A chance node is a way to introduce uncertainty introduce uncertainty into a game beyond the uncertainty created by the players' into a game beyond the uncertainty created by the players' use of mixed strategy. use of mixed strategy. An example might be different states of states of naturenature that may or may not be resolved before the players make that may or may not be resolved before the players make their choices. their choices.



Nash Bargaining Solutions

1 Bargaining SolutionIn a transaction when the seller and the buyer value a product differently, a surplus is created. A A bargainingbargaining solution is then a way in which buyers and solution is then a way in which buyers and sellers sellers agree to divide the surplus.agree to divide the surplus. For example, consider a house made by a builder A. It costed him 10 . A potential buyer is interested in the house and values it at 20. This transaction can generate a surplus of 10 . The builder and the buyer now need to trade at a price. The buyer may know that the cost is less than 20 and the seller may know that the value is greater than 10 . When trade is feasible ,they need to agree at a price. Both try to maximize their surplus. Buyer can not buy it for less than10 , while the seller can not sell it for more than 20 . They bargain on the price, and either trade or dismiss. Trade would result in the generation of surplus, whereas no surplus is created in case of no-trade. Bargaining Solution is defined as,

F : (X,d) → S,



where X R2 and S,d R2 . X represents the utilities of the players in the set of possible bargaining agreements. d represents the point of disagreement. In the above example, price [10,20], bargaining set is simply x + y 10, x 0, y 0. A point (x,y) in the bargaining set represents the case, when seller gets a surplus of x, and buyer gets a surplus of y, i.e. seller sells the house at 10 + x and the buyer pays 20 y. Assumption Bargainging Set X is convex and bounded. Figure 1: Bargaining Set

F : (X,d) S,



2 Pareto Optimality

A Pareto Optimal solution is one in which none of the players can increase their payoff without decreasing the payoff of at least one of the other players. where ui(w) is the utility function for player i at outcome w. All points on the boundary of the Bargaining Set are Pareto Optimal solutions. In a bargaining situation, players would like to settle at a pareto optimal outcome, because if they settle at an outcome which is not pareto optimal, then there exists another outcome where at least one player is better off without hurting the interest of the other players. Pareto optimal solutions are not unique in most of the cases. Example. In the earlier example, x + y = 10 is a pareto optimal frontier.

3 Properties of a Bargaining SolutionNash gave four axioms that any bargaining solution should satisfy. 1-invariant to affine transformations. 2-Pareto optimality. 3-Independence from Irrelevant Alternatives. 3-Symmetry


Invariant to affine transformations An affine transformation Ab : R

2 R2 is defined by a matrix A, and a vector b of the

following form:

A = [

0]

2

b = [ ]

Now the transformation of X can be defined as:

A bargaining solution is invariant to an affine transformation iff A, b, if

then

1

0

1

2

Ab(x) = Ax + b

F(X,d) = S

F(Ab(X),Ab(d)) = Ab(S)


Nash Bargaining Solutions • Pareto OptimalityF(X,d) should be a Pareto optimal solution. Any bargaining solution should be better off than the disagreement point. •Independent from Irrelevant AlternativesIf S is the Nash bargaining solution for a bargaining set X then for any subset Y of X containing S, S continues to be the Nash Bargaining Solution. This axiom of Nash is slightly controversialcontroversial unlike the previous two axioms, since more alternatives give you better bargaining power. However, this can be intuitively justified, by the following argument: Let us say that the set Y has a NBS S' and S be another NBS of X (refer figure 2). Now S Y, S Y and S X, S X . In both the bargaining sets X and Y, both the options S, S' are available to the players. They should be expected to settle to the same outcomes. The presence of irrelevant alternatives in X should not influence the bargaining solution. Formally, if

and F(X,d) = S

Y X , S Y , d Y,

F(Y,d) = S

Figure 2: Independence from Irrelevant Alternatives



Symmetry

The principle of symmetry says that symmetric utility functions should ensure symmetric payoffs. Payoff should not discriminate between the identities of the players. It should only depend on their payoff functions. Put simply, symmetry implies the bargaining solution for region X = x + y 1, x 0, y 0, d = (0,0) , should be (1/2,1/2) as shown in figure 3. If both players have the same utility functions, then symmetry demands that both get equal payoffs.

Figure 3: Symmetry


Problems ;

H & Q problem no. 8-6 ;

Consumers distributed uniformly along a straight line road are the potential market for two duopolists whose decision problem is where to locate their sale offices . Demand is completely inelastic , and consumers will purchase from wherever sales office is nearer. Assume that the road is 4 miles long , and that for simplicity , each firm has exactly five possible strategies ; it may locate itself at either end or at , 1 – mile , 2 – mile , 3 –mile , markets . Let the payoffs to the duopolists be their respective market shares.

a- Is this a zero - sum ( constant – sum ) game ?

b- what iis the payoff matrix ?

c- What are the optimal strategies for the duopolies?


Solution ;

consumers distribution uniformly . In each mile 25 percent of them is located .

0 25 50 75 100 1 2 3 4

I II

1-mile

2-mile

3-mile

4-mile

0-mile

0-mile

1-mile 2-mile 3-mile 4-mile

0.5 0.375

0.625

0.875

0.1250.5

0.875

0.75

0.625

0.5

0.875

0.125

0.5

0.625

0.8750.5

0.375

0.375

0.625

0.25

0.5

0.5

0.375

0.25

0.375

0.5 0.375

0.625 0.75

0.125

0.5

0.5

0.5

0.625

0.125

MIN

MAX

First duopolist matrix payoffMAX MIN I = MIN MAX II = 0.5


problem no. 8-7 H &Q .

Show that the feasible utility region for mixed strategies in figure 8-3 is ABCD if the duopolists have two pure strategies each as stated in the discussion of Fig 8-3 .

Duopolist II

Leader Follower

Leader (200,250) (1000,200)

Duopolist I

Follower (150,950) (800 ,800 )

U2

U1

A

B

C

D

T

E wwW = ( UW = ( U11 – U – U00

11 )( U )( U22 – U – U0022 ) )

UU0011

UU0022A I will employ a pure strategy as I will employ a pure strategy as

follower and II will employ a follower and II will employ a mixed strategy with probability mixed strategy with probability of leadership equal to EB/BDof leadership equal to EB/BD

game theoryh @ q ch 81 introduction what economists call game theory psychologists call the theory...

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